Consecutive Quadratic Residues And Quadratic Nonresidue Modulo $p$

小素数k次幂剩余与非剩余在算术级数中的情况

Small Prime $k$th Power Residues and Nonresidues in Arithmetic Progressions

本文研究了在算术级数中存在小素数k次幂剩余和非剩余的条件，并推广了二次剩余的结果。

最小连续二次非剩余对

Least Consecutive Pair of Quadratic Nonresidues

研究大素数域中最小连续二次非剩余对的上界，证明其不超过 O((log p)^2 (log log p)^3)。

关于勾股三元组结构的两个定理及丢番图推论

Two Theorems on the structure of Pythagorean triples and some diophantine consequences

尽管本文实际上证明了四个定理，但其中两个是主要的定理，即定理1和定理3。在定理1中，我们证明了如果a和be是满足一定的二次剩余条件的无平方的正整数，则不存在一个边长等于a乘一个整数平方，而另一个边长等于b乘一个完满平方的本原毕达哥拉斯三角形。所有这样的对（a，b）的族在其描述中稍微复杂。该族的一个子类由数对（a，b）组成，a全等于1，b全等于5模8; a和b都是素数，a是b的二次非剩余（因此根据二次互反律，b也是a的非剩余）。定理3在性质上类似，但其假设不那么复杂。如果p和q都是素数，且都与1模4全等，且其中一个是另一个的二次非剩余，则丢番图方程p^2x^4 + q^2y^4 = z^2在满足（px，qy）=1的正整数x，y和z中没有解。

Maximum distance between consecutive primes and other related questions

The paper considers the asymptotic of the ratio of the number of primes not exceeding the primorial and the number of residues in the reduced system of residues for the given primorial. We study the relationship between asymptotic lower bounds for the values of the Jacobstal function and the maximum distance between successive primes. One algorithm for computing the Jacobstal function is given. The substantiation of the conjectures about the upper estimate of the maximum distance between successive prime numbers is given.

Simultaneous Elements Of Prescribed Multiplicative Orders

Let $u\ne \pm 1$, and $v\ne \pm 1$ be a pair of fixed relatively prime squarefree integers, and let $d\geq 1$, and $e \geq1$ be a pair of fixed integers. It is shown that there are infinitely many primes $p\geq 2$ such that $u$ and $v$ have simultaneous prescribed multiplicative orders $\text{ord}_pu=(p-1)/d$ and $\text{ord}_pv=(p-1)/e$ respectively, unconditionally. In particular, a squarefree odd integer $u>2$ and $v=2$ are simultaneous primitive roots and quadratic residues (or quadratic nonresidues) modulo $p$ for infinitely many primes $p$, unconditionally.

连续素数分布中的预期偏差

Expected biases in the distribution of consecutive primes

本文基于埃拉托斯特尼筛法各阶段的间隔种群演化，解释了Lemke Oliver和Soundararajan观察到的素数间隔模10分布偏差，指出该偏差源于小间隔的快速起步和剩余类分配不均，且最终是暂时的。